The zeta function of the infinite grid

The zeta function of a graph is the generating function counting certain prime loops in the graph. For the infinite grid, the zeta function is given by a double integral.

In this talk, I'll give the background behind the zeta function and some connections to the classical Ising model of a ferromagnet. Then, using theta functions, I'll discuss the analytic continuation of the zeta function to a Riemann surface.